Fulde-Ferrell-Larkin-Ovchinnikov state in bilayer dipolar systems

TL;DR

This paper investigates the phase diagram of bilayer fermionic polar molecules, revealing how long-range dipolar interactions promote the FFLO state with stripe order under density imbalance.

Contribution

It demonstrates that long-range dipolar interactions enhance the FFLO phase and enable mixed s-wave and p-wave pairing in bilayer systems.

Findings

FFLO state appears with stripe order at high imbalance

Long-range dipolar interactions expand the FFLO regime

Mixed s-wave and p-wave pairing is possible due to dipolar interactions

Abstract

We study the phase diagram of fermionic polar molecules in a bilayer system, with an imbalance of molecular densities of the layers. For the imbalance exceeding a critical value the system undergoes a transition from the uniform interlayer superfluid to the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with a stripe structure, and at sufficiently large imbalance a transition from the FFLO to normal phase. Compared to the case of contact interactions, the FFLO regime is enhanced by the long-range character of the interlayer dipolar interaction, which can combine the s-wave and p-wave pairing in the order parameter.

Equations

$$V(\mathbf{r})=d^2(r^2-2{\lambda }^2)/(r^2+{\lambda }^2{)}^{5/2},$$

$$V_{\mathbf{k}{\mathbf{k}}^{\prime }}=\int d\mathbf{r}V(\mathbf{r})e^{i({\mathbf{k}}^{\prime }-\mathbf{k})\cdot \mathbf{r}}=-2\pi d^2\kappa e^{-\kappa \lambda },$$

$$H_I=\sum _{\mathbf{k}{\mathbf{k}}^{\prime }\mathbf{q}}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}{c}_{-{\mathbf{k}}^{\prime }+\frac{\mathbf{q}}{2},\uparrow }^{†}{c}_{{\mathbf{k}}^{\prime }+\frac{\mathbf{q}}{2},\downarrow }^{†}{c}_{\mathbf{k}+\frac{\mathbf{q}}{2},\downarrow }{c}_{-\mathbf{k}+\frac{\mathbf{q}}{2},\uparrow }$$

$${\Delta }_{\mathbf{k}}(\mathbf{R})=\sum _{n=1}^{N_Q}{\Delta }_{\mathbf{k}{\mathbf{Q}}_n}{e}^{i{\mathbf{Q}}_n\cdot \mathbf{R}},$$

$$G_{\sigma {\sigma }^{\prime }}({\mathbf{k}}_1,{\mathbf{k}}_2;i{\omega }_n)={\delta }_{\sigma {\sigma }^{\prime }}{\delta }_{{\mathbf{k}}_1,{\mathbf{k}}_2}$$

$$F_{\sigma {\sigma }^{\prime }}^{†}({\mathbf{k}}_1,{\mathbf{k}}_2;i{\omega }_n)=\frac{-{\sum }_{m=1}^{N_Q}{\Delta }_{{\mathbf{k}}_1,{\mathbf{Q}}_m}^{†}{\delta }_{{\mathbf{k}}_1+{\mathbf{Q}}_m,{\mathbf{k}}_2}}{i{\omega }_n+{\xi }_{{-\mathbf{k}}_1\sigma }}$$

$${\Delta }_{\mathbf{k}\mathbf{Q}}^{\ast }=-T\sum _{n,{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}{F}_{\uparrow \downarrow }^{†}({\mathbf{k}}^{\prime }-\frac{\mathbf{Q}}{2},{\mathbf{k}}^{\prime }+\frac{\mathbf{Q}}{2};{\omega }_n).$$

$$F_2=\sum {\Delta }_{\mathbf{k}\mathbf{q}}(V^{-1}{)}_{\mathbf{k}{\mathbf{k}}^{\prime }}{\Delta }_{{\mathbf{k}}^{\prime }\mathbf{q}}^{\ast }-T\sum |{\Delta }_{\mathbf{k}\mathbf{q}}{|}^2{G}_{+}(\mathbf{k}+\frac{\mathbf{q}}{2},{\omega }_n)G_{-}(-\mathbf{k}+\frac{\mathbf{q}}{2},-{\omega }_n),$$

$$F_4=\frac{T}{2}\sum _{\{{\mathbf{q}}_j\},\mathbf{k}n}{\Delta }_{\mathbf{k}{\mathbf{q}}_1}^{\ast }{\Delta }_{\mathbf{k}{\mathbf{q}}_2}^{\ast }{\Delta }_{\mathbf{k}{\mathbf{q}}_3}{\Delta }_{\mathbf{k}{\mathbf{q}}_4}{G}_{+}(\mathbf{k}+{\mathbf{q}}_1,{\omega }_n)G_{-}(-\mathbf{k}+{\mathbf{q}}_3-{\mathbf{q}}_1,-{\omega }_n)G_{+}(\mathbf{k}+{\mathbf{q}}_4,{\omega }_n)G_{-}(-\mathbf{k},-{\omega }_n){\delta }_{{\mathbf{q}}_1+{\mathbf{q}}_2,{\mathbf{q}}_3+{\mathbf{q}}_4},$$

$$G_{\pm }(\mathbf{k},{\omega }_n)=\frac{1}{i{\omega }_n-{\xi }_{\mathbf{k}}\mp h},$$

$$-\ln \frac{T_{0,s}}{T_{0p}}+\mathrm{Re}\left[\Psi \left(\frac{1}{2}+i\frac{h^{\ast }}{2\pi T^{\ast }}\right)-\Psi \left(\frac{1}{2}\right)\right]=0;$$

$$-\mathrm{Re}\left[{\Psi }^{\prime \prime }\left(\frac{1}{2}+i\frac{h^{\ast }}{2\pi T^{\ast }}\right)\right]=\frac{{\left(\mathrm{Im}{\Psi }^{\prime }\left(1/2+i{h}^{\ast }/2\pi T^{\ast }\right)\right)}^2}{\ln (T_{0,s}/T_{0,p})},$$

Full text

Exotic many-body quantum states in population imbalanced spin-1/2 Fermi systems attract a great deal of interest, to a large extent due to expected non-conventional transport properties. Among these states, the most actively studied is the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase FF ; LO , in which Cooper pairs have finite momenta and the order parameter shows a lattice structure on top of a uniform background. Theoretical studies of the FFLO phase in condensed matter are lasting for decades Bulaevski ; Buzdin ; BuzdinPLA97 ; Casalbuoni ; FFLOHeavyF , and rapid developments in the field of ultracold quantum gases stimulated the studies of this phase in two-component imbalanced Fermi gases LeoRPP ; GubbelsPhysRep ; CombescotEPL ; KetterleRev ; LeoPRA84 ; MuellerPRA79 ; SheehyPRA85 . However, experimental verification of the existence of the FFLO state is still in progress BianchiPRL03 ; RadovenNat03 ; Ketterle06 ; Ketterle07 ; HuletScience06 ; KetterleRev . In ultracold gases the search for the FFLO phase is actively pursued for strongly interacting fermions, where one has a crossover from Bardeen-Cooper-Shrieffer (BCS) superfluid to Bose-Einstein condensate of weakly bound molecules Pitaev ; BlochRMP ; Levin ; Ketterle06 ; Ketterle07 ; HuletScience06 , and the population imbalance is expected to lead to the FFLO state MuellerPRA79 ; LeoPRA84 ; SheehyPRA85 .

Recent advances in creating ultracold polar molecules Ni ; YeNJP interacting with each other via long-range anisotropic dipole-dipole forces open fascinating prospects for many-body physics BaranovPhysRep ; ZollerChemRev . A variety of novel many-body states was proposed for fermionic dipoles in two dimensions (2D) Taylor ; Cooper2009 ; Gora ; Pikovski2010 ; Potter ; Sun ; Miyakawa ; DasSarmaFL ; Ronen ; BaranovSieb ; ZinnerDW ; Parish ; Babadi ; Lu1 ; Lu2 ; Giorgini , including interlayer superfluids with the BCS-BEC crossover in a bilayer geometry Pikovski2010 ; Ronen ; ZinnerDW . Importantly, in 2D the decay of polar molecules due to ultracold chemical reactions Jin ; Jin2 can be suppressed by orienting the dipoles perpendicularly to the plane, which induces a strong intermolecular repulsion Bohn1 ; BaranovM ; YeM . Together with possible experiments with non-reactive polar molecules ZwierleinW ; GrimmT , this forms a promising path towards new many-body quantum states.

In this letter we predict wide possibilities for creating the FFLO phase of polar molecules in a bilayer geometry, with a finite imbalance of molecular densities of the layers. Cooper pairs are formed by molecules belonging to different layers due to the interlayer dipole-dipole interaction, and the most favoured is the FFLO state with a stripe structure of the order parameter. Remarkably, the FFLO regime of this interlayer superfluid is enhanced by the long-range character of the dipolar interaction, which in an imbalanced system may lead to Cooper pairs representing superpositions of contributions of various partial waves. Our work thus opens a new direction to investigate novel superfluids of fermionic particles with population imbalance.

Interlayer interaction and order parameters: We consider identical fermionic polar molecules in a bilayer geometry, oriented perpendicularly to the layers by an electric field (see Fig. 1). The interlayer dipole-dipole interaction is partially attractive and it may lead to interlayer superfluid pairing Pikovski2010 ; Potter ; Ronen ; ZinnerDW . The inlayer interaction is purely repulsive and it only renormalizes the chemical potential Pikovski2010 ; ZinnerDW , so that below we omit this interaction. We assume that molecular densities in the layers are different from each other, but in each layer the Fermi energy greatly exceeds the binding energy of interlayer dimers (bound states of dipoles belonging to different layers Pikovski2010 ; Ronen ; Klawunn ). Thus, the only effect of the interlayer dipole-dipole interaction will be the fermionic superfluid pairing, which we consider in the weakly interacting BCS regime.

The interlayer dipole-dipole interaction potential and its Fourier transform are given by

[TABLE]

where $\kappa\equiv|{\bf k}-{\bf k}^{\prime}|$, $d$ is the effective dipole moment of a molecule, and ${\bf r}$ is the inplane separation between two dipoles.The pairing potential can be expanded in a series in angular momenta $V_{{\bf k}{\bf k}^{\prime}}=\sum_{l}V_{l}(k,k^{\prime})e^{i(\phi-\phi^{\prime})l}$. The leading part of the scattering (interaction) amplitude can be obtained in the Born approximation. It is important that in contrast to contact interactions, the dipolar (interlayer) interaction amplitude with $\l|>0$, and especially the $p$-wave amplitude, can be comparable with the $s$-wave amplitude.

After omitting the inlayer interaction the system maps onto spin-1/2 fermions with intercomponent dipolar interaction. The Hamiltonian reads $H=H_{0}+H_{I}$, where $H_{0}=\sum_{\mathbf{k},\sigma}\xi_{\mathbf{k},\sigma}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}$ is the kinetic energy, and

[TABLE]

is the interaction energy. Here $c_{\mathbf{k}\sigma}$ are fermionic field operators, $\sigma=\uparrow,\downarrow$, stands for the layer index, $\xi_{\mathbf{k}\sigma}=k^{2}/2m-\mu_{\sigma}$ (hereinafter we put $\hbar=1$), and $\mu_{\sigma}=\mu\pm h$ are chemical potentials of the layers. When the densities in the layers are not equal to each other, there is a difference in the Fermi momenta of the two (pseudo)spin states, $\delta k_{F}\equiv|k_{F,\uparrow}-k_{F,\downarrow}|\neq 0$, and the effective magnetic field $h\neq 0$.

Relying on Eq.(3) we define the pairing gap function as $\Delta_{\mathbf{kQ}}\equiv\sum_{\mathbf{k}^{\prime}}V_{\mathbf{kk^{\prime}}}\langle c_{\mathbf{k}^{\prime}+\frac{\mathbf{Q}}{2},\downarrow}c_{-\mathbf{k}^{\prime}+\frac{\mathbf{Q}}{2},\uparrow}\rangle$, where ${\bf k}^{\prime}$ is the relative momentum of two paired fermions, and ${\bf Q}$ is their center-of-mass (CM) momentum. In the coordinate space, the order parameter is then consisting of pairing wavefunctions for several CM momenta:

[TABLE]

where $\mathbf{R}$ is the CM position of the Cooper pair, and $\mathbf{Q}_{n}$ is the CM momentum involved in the order parameter. Below we consider several symmetries of the order parameter: uniform superfluid ($N_{Q}=1$ and ${\bf Q}_{1}=0$), plane wave FFLO ($N_{Q}=1$ and ${\bf Q}_{1}=q\hat{x}$, where $\hat{x}$ is a unit vector in the $x$ direction), stripe FFLO ($N_{Q}=2$ and ${\bf Q}_{1,2}=\pm q\hat{x}$), and triangular state ($N_{Q}=3$ and three ${\bf Q}_{n}$ vectors have the same amplitude, with $2\pi/3$ difference in their orientation). Close to the FFLO-normal phase boundary we also consider square and hexagonal FFLO structures.

The finite-temperature normal and anomalous Green functions $G_{\sigma\sigma^{\prime}}$ and $F^{\dagger}_{\sigma\sigma^{\prime}}$ are found from the Gor’kov equations Gorkov ; MahanMineev (see Supplemental Material for details):

[TABLE]

[TABLE]

where $\omega_{n}=(2n+1)\pi T$ are Matsubara frequencies, and $T$ is the temperature. The gap equation can then be obtained self-consistently:

[TABLE]

We thus identify the Gor’kov equations for the Green functions and the gap equations for $\Delta^{*}_{{\bf k}{\bf Q}}$ and $\Delta_{{\bf k}{\bf Q}}$ as self-consistent Gor’kov equations.

The plane wave, stripe, triangular, square, and hexagonal phases break the rotational symmetry and have an anisotropic gap in the momentum space. In principle, such gap function can be measured using Bragg spectroscopy by exciting particles with a finite momentum.

In the following, we first solve the self-consistent Gor’kov equations by assuming a certain gap function symmetry (uniform, plane wave, stripe, or triangular) and then determine the phase diagram by comparing the obtained free energies of these candidates. The derivation of equations (5)-(9) and a detailed presentation of the numerical procedure of obtaining the phase diagrams at zero and finite temperatures are contained in the Supplemental Material.

Near the phase transition line (superfluid – normal state), where the order parameter is small, we may use the Ginzburg-Landau free energy functional $F=F_{2}+F_{4}+...$, with

[TABLE]

[TABLE]

where the Green’s functions of the normal state are

[TABLE]

and $V^{-1}$ is the inverse matrix of $V_{{\bf k}{\bf k}^{\prime}}$. The term $F_{4}$ is necessary to find the minimum energy configuration, whereas $F_{2}$ determines the tricritical point.

Zero temperature: At zero field ($h=0$) the ground state is a uniform superfluid with the order parameter on the Fermi surface $\Delta_{0}(k_{F})\equiv\Delta_{0}$, where $k_{F}=\sqrt{2m\mu}$ is the Fermi momentum at $h=0$. For a given interaction strength the FFLO phase emerges at a critical value $h_{c1}$, and our calculations show that this is a stripe phase. For sufficiently large field $h_{c2}$ the ground state becomes normal. Note that the stripe FFLO phase is clearly the ground state at $h$ that are lower than $h_{c2}$ by a few percent (see Supplemental Material). The dependence of $h_{c1}$, $h_{c2}$ on the parameter $k_{F}\lambda$ displayed in Fig. 2 shows that for $k_{F}\lambda>1$ the FFLO region is significantly wider than in the case of contact interaction, where the FFLO phase emerges for $\Delta_{0}>h>\Delta_{0}/\sqrt{2}\approx 0.707\Delta_{0}$. Here we arrive at a very important point. For $k_{F}\lambda<1.055$ the $p$-wave interaction on the Fermi surface is repulsive and interlayer Cooper pairs only contain the contribution of an attractive $s$-wave interaction. However, if $k_{F}\lambda>1.055$, then the $p$-wave interaction $V_{1}(k_{F},k_{F})$ becomes also attractive and Copper pairs are already composed of both $s$-wave and $p$-wave contributions. This makes the modulus of the order parameter larger and requires a higher field $h_{c2}$ to destroy superfluidity and get to the normal state (see Supplemental Material).

Finite temperature phase diagram: At finite temperatures the equilibrium phases are determined by comparing the free energies of the uniform superfluid, FFLO, and the normal state. In Fig. 3 the phase diagram is presented in terms of $T/T_{0,s}$ and $h/\Delta_{0}$, where $T_{0,s}$ is the transition temperature at $h=0$ (provided by the $s$-wave pairing). With increasing temperature, the critical field $h_{c1}$ for the transition from the uniform superfluid to FFLO phase decreases. So does the imbalance for the transition from the FFLO to the normal state. The three phases (SF, FFLO, and normal) merge at the tricritical point $T^{*}$. For $k_{F}\lambda<1$ we have $T^{*}\approx 0.56T_{0,s}$ like in the case of contact interactions. However, for $k_{F}\lambda>1$ the $p$-wave pairing contribution to the order parameter comes into play and the ratio $T^{*}/T_{0,s}$ increases. Already for $k_{f}\lambda=2.2$ we obtain $T^{*}\simeq 0.62T_{0,s}$ and somewhat wider FFLO region than in the case of contact interactions (see Fig. 3).

The FFLO phase is mostly the Larkin-Ovchinnikov stripe state. In the temperature interval from $0.02T_{0,s}$ to $0.3T_{0,s}$ the stripe phase has the lowest free energy for $h$ lower than $h_{c2}$ by more than $5\%$. For $h$ closer to $h_{c2}$ our numerical calculations based on the Ginzburg-Landau functional show that the equilibrium state is a triangular FFLO (it is also recovered from the self-consistent Gor’kov equations at $h$ close to $0.95h_{c2}$), which with decreasing temperature becomes a square and then hexagonal FFLO (see Supplemental Material). This sequence of FFLO states is similar to that observed for contact interactions Shimahara ; FFLOHeavyF ; CombescotEPL .

The structure of the phase diagram is similar to that in the case of contact interactions LO ; BuzdinPLA97 ; Shimahara . However, in our case the FFLO region significantly depends on the parameters, so that $T^{*}/T_{0,s}$ and $h^{*}/\Delta_{0}$ are not universal numbers as they are for contact interactions where $T^{*}\approx 0.56T_{0,s}$ and the corresponding critical imbalance is $h^{*}\approx 0.6\Delta_{0}$ BuzdinPLA97 ; Shimahara ).

The tricritical temperature can be determined analytically (see Supplemental Material for details):

[TABLE]

[TABLE]

where $T_{0,p}$ is the critical temperature of superfluid transition for the $p$-wave pairing at $h=0$, and $\Psi$ is the digamma function. In the limit $T_{0,p}\rightarrow 0$ Eqs (11) and (12) give the known result for the contact interaction, which is specified in the previous paragraph. These equations also reproduce numerical results of Figs. 3 and 4. The maximum tricritical temperature $T^{*}\to T_{0,s}$ is reached in the limit $k_{F}\lambda\gg 1$, where $T_{0,p}$ is close to $T_{0,s}$. However, the critical temperature $T_{0,s}$ decreases with increasing $k_{F}\lambda$. For $k_{F}\lambda\simeq 2$, where one can still hope to achieve $T_{0,s}$ on the level of nanokelvins (see below), we obtain $T^{*}\approx 0.62T_{0,s}$.

The lattice wavevector $|{\bf Q}_{m}|$ behaves as $Q\sim\sqrt{T_{c}^{*}-T}$ near the tricritical point and vanishes at $T=T_{c}^{*}$ (see Supplemental Material).

As is already said above, the equilibrium FFLO phase in our case is the stripe state. Like in the case of contact interactions burkh ; Shimahara , the transition from the uniform superfluid to FFLO state is of the first order, whereas the transition from the FFLO to normal phase is of the second order. At 2D densities $\sim 10^{9}\text{cm}^{-2}$ the Fermi energy for weakly reactive NaLi molecules or non-reactive NaK molecules is $\epsilon_{F}\sim 1\mu$K, and for the interlayer spacing $200$ nm we have $k_{F}\lambda\simeq 2$. Then, on approach to the strongly interacting regime with $k_{F}r_{*}$ slightly exceeding unity the superfluid transition temperature will be up to $10$ nK. Larger values of $k_{F}\lambda$ and, hence, higher ratios $T^{*}/T_{0,s}$ would require $k_{F}r_{*}$ significantly larger than unity, so that the inlayer interaction becomes very strong driving the system far away from the dilute regime. This case is beyond the scope of the present paper.

Conclusions: We used both a theoretical field approach based on the Gor’kov equations and the theory of phase transitions based on the Ginzburg-Landau free energy functional to study possible FFLO phases in a bilayer system of fermionic polar molecules with a finite imbalance of molecular densities of the layers. Our work demonstrates the importance of the long-range character of the dipole-dipole interaction, which can combine the $s$-wave and $p$-wave pairing in the order parameter and enhance the FFLO regime. The observation of this FFLO state is feasible for non-reactive NaK molecules or weakly reactive NaLi molecules at temperatures $\sim 10\text{nK}$.

We are grateful to Alexander Buzdin for fruitful discussions. We acknowledge support from MOST and NCTS in Taiwan. The research leading to these results has received funding from the European Research Council under European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 341197).